Menu Close

Find-the-max-and-min-of-function-a-bsin-x-b-asin-x-where-b-gt-a-gt-0-in-the-interval-0-x-2pi-sketch-a-4-and-b-5-




Question Number 88040 by peter frank last updated on 08/Apr/20
Find the max and min  of function  ((a+bsin x)/(b+asin x))  where b>a>0 in the   interval 0≤x≤2π.sketch  a=4 and b=5
$${Find}\:{the}\:{max}\:{and}\:{min} \\ $$$${of}\:{function} \\ $$$$\frac{{a}+{b}\mathrm{sin}\:{x}}{{b}+{a}\mathrm{sin}\:{x}} \\ $$$${where}\:{b}>{a}>\mathrm{0}\:{in}\:{the}\: \\ $$$${interval}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\pi.{sketch} \\ $$$${a}=\mathrm{4}\:{and}\:{b}=\mathrm{5} \\ $$
Commented by john santu last updated on 08/Apr/20
f(x) = ((bsin x+a)/(asin x+b))  f′(x) = (((b^2 −a^2 )cos x)/((asin x+b)^2 )) = 0  (b^2 −a^2 ) ≠0 , then cos x = 0  cos x = 0  { ((x_1 =(π/2))),((x _2 = ((3π)/2))) :}  f(x_1 ) = ((b+a)/(a+b)) = 1 ←max  f(x_2 ) = ((a−b)/(b−a)) = −1 ←min
$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{bsin}\:\mathrm{x}+\mathrm{a}}{\mathrm{asin}\:\mathrm{x}+\mathrm{b}} \\ $$$$\mathrm{f}'\left(\mathrm{x}\right)\:=\:\frac{\left(\mathrm{b}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} \right)\mathrm{cos}\:\mathrm{x}}{\left(\mathrm{asin}\:\mathrm{x}+\mathrm{b}\right)^{\mathrm{2}} }\:=\:\mathrm{0} \\ $$$$\left(\mathrm{b}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} \right)\:\neq\mathrm{0}\:,\:\mathrm{then}\:\mathrm{cos}\:\mathrm{x}\:=\:\mathrm{0} \\ $$$$\mathrm{cos}\:\mathrm{x}\:=\:\mathrm{0}\:\begin{cases}{\mathrm{x}_{\mathrm{1}} =\frac{\pi}{\mathrm{2}}}\\{\mathrm{x}\:_{\mathrm{2}} =\:\frac{\mathrm{3}\pi}{\mathrm{2}}}\end{cases} \\ $$$$\mathrm{f}\left(\mathrm{x}_{\mathrm{1}} \right)\:=\:\frac{\mathrm{b}+\mathrm{a}}{\mathrm{a}+\mathrm{b}}\:=\:\mathrm{1}\:\leftarrow\mathrm{max} \\ $$$$\mathrm{f}\left(\mathrm{x}_{\mathrm{2}} \right)\:=\:\frac{\mathrm{a}−\mathrm{b}}{\mathrm{b}−\mathrm{a}}\:=\:−\mathrm{1}\:\leftarrow\mathrm{min} \\ $$
Commented by peter frank last updated on 15/Apr/20
thank you
$${thank}\:{you} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *